Distribution of Maximun Likelihood Estimator

[Tranquillity]

Hey guys how are you? I have the following question:

Let X1,X2,...,Xn be a random sample from a Pareto distribution having pdf
f(x|b)= (a*b^a)/x^(a+1) where x>=b (1)

Determine the maximum likelihood estimator for b, say b' on (0,infinity) and by considering P(b'>x) or otherwise show that b' has the Pareto distribution with pdf given by (1) but with a replaced by an.


My attempt: I found the MLE as b'=min Xi where 1<=i<=n, since our pdf is monotonically increasing w.r.t b.

After that I know how to find the asymptotic distribution of the MLE using the formula including the expected information but then we say that MLE follows a normal distribution for large n.

How do I show that the MLE follows a Pareto distribution in this case? I am so struggled, any help would be much appreciated!

P.S The hint tells us to consider P(b'>x) but how can I find P(min Xi >x) and why should it help me?

 

[bpet]

Hint: min Xi > x if and only if X1>x and X2>x etc.

If you write the distribution's cdf instead of pdf it'll be much easier.

 

[Tranquillity]

Let say P(b'>x) = P(Xmin>x)= P(X1>x, X2>x,...,Xn>x)= P(X1>x) * P(X2>x) * ...*P(Xn>x)= [P(X1>x)]^n (since they are independent) But then what is P(X1>x)? Have I used correctly the independence?